Air carrier pilot reported a malfunction of a squat switch after putting the gear up after takeoff. The crew advised ATC and returned to the field.

Narrative

We were told that the aircraft had some landing gear work done and that it did not have a heavy check done. It was the first flight of the day for this aircraft so I did my normal preflight walk around first flight of day checks and specifically looked over the gear thoroughly. I saw nothing abnormal on the preflight walk around. The Captain also did a walk around and didn't notice anything abnormal. After getting our clearance we requested to hold near the runway to balance the fuel. We had more than 500lbs center tank but not 4400lbs in each wing tank. After complying with this limitation we requested takeoff clearance. We were told left turn on course to ZZZ1 and climb to 6000. The crosswind was strong so a lot of wind correction was used but everything was normal during the initial takeoff phase. As soon as the Captain called positive rate I called gear up/speed mode. Almost immediately we get a STEERING INOP caution msg. A few seconds later we get a WOW INPUT caution msg and also notice a WOW Output fail status msg. The Captain immediately lets ATC know we have an issue and we need to run some checklists. They ask do we want to continue on course and for now we tell them sure but we also want to stay in the area. So I keep us at 200 IAS. The first QRH for STEERING INOP was performed and it did not fix the issue. WOW INPUT QRH was then performed and it did not fix the issue either. Dispatch was contacted advising of the situation and we requested with dispatch to continue to ZZZ if they agreed because the QRH said land on the longest runway available with a low crosswind because we may not have steering upon landing. While waiting for Dispatch to respond the Captain and I agreed out of safety to advise ATC. ATC was notified and then requested souls on board/fuel on board/nature of the problem. They were advised all of that. Dispatch ACARS us back and agreed that proceeding to ZZZ but try to cycle the gear before landing in ZZZ. So ATC was then advised we were proceeding to ZZZ and they climbed us up to 17000. Once leveling off at 17000 the Captain and I agreed we would transfer pilot flying (PF)/pilot monitoring (PM) duties once we got closer to ZZZ. We were told to descend at pilots discretion to 8000 for RNAV Runway XX ZZZ. At this point we slowed down and the gear was requested down. I put the gear down and the reset was attempted again. It fixed the issues and the msgs went away. I then advised ATC issues were solved. They asked did we still need the fire trucks waiting and I told ATC negative. The Captain had a smooth landing. ATC asked did we require assistance off the runway and I told them STBY to make sure we were able to turn off the runway onto the taxiway. Once we did I told them no assistance needed. We asked for taxi to [the FBO]. We taxied there and the fire trucks followed us to parking. On the post flight walk around nothing abnormal was noticed. The plane was written up for Maintenance after we parked. We also filled out some paperwork the emergency service crew needed. I then sat in with my Captain for a phone debrief and Assistant Chief Pilot (ACP)/Maintenance/Dispatcher etc. were all involved on the call just asking for an overview of everything. Overall I think we did a good job. Followed SOP; ran the QRH procedures; and ultimately we were fine and the aircraft was landed safely. Good CRM between the Captain and I.The aircraft was in a Maintenance facility and specifically had the gear worked on which is the issue we encountered right after takeoff rotation. Maybe the Maintenance crew after working on the plane can cycle the gear a few times to see if they get any abnormalities/msgs before the aircraft is scheduled to fly. Also no where in the QRH procedures does it say to cycle the gear for the caution msgs we received. Maybe that can be added to the QRH procedures so for the future crews can attempt this to see if it resolves the issue.